Mathematics

Find $$\int \dfrac { d x } { x ^ { 2 } - a ^ { 2 } }$$  and hence evaluate $$\int \dfrac { d x } { x ^ { 2 } - 25 }$$.


SOLUTION
$$\displaystyle\int{\dfrac{dx}{{x}^{2}-{a}^{2}}}$$

$$=\displaystyle\int{\dfrac{dx}{\left(x-a\right)\left(x+a\right)}}$$

$$=\dfrac{1}{2a}\displaystyle\int{\dfrac{\left(\left(x+a\right)-\left(x-a\right)\right)dx}{\left(x-a\right)\left(x+a\right)}}$$

$$=\dfrac{1}{2a}\displaystyle\int{\dfrac{\left(x+a\right)dx}{\left(x-a\right)\left(x+a\right)}}-\displaystyle\int{\dfrac{\left(x-a\right)dx}{\left(x-a\right)\left(x+a\right)}}$$

$$=\dfrac{1}{2a}\displaystyle\int{\dfrac{dx}{\left(x-a\right)}}-\displaystyle\int{\dfrac{dx}{\left(x+a\right)}}$$

$$=\dfrac{1}{2a}\left[\log{\left|x-a\right|}-\log{\left|x+a\right|}\right]+c$$

$$=\dfrac{1}{2a}\log{\left|\dfrac{x-a}{x+a}\right|}+c$$

Now $$\displaystyle\int{\dfrac{dx}{{x}^{2}-25}}$$

$$=\displaystyle\int{\dfrac{dx}{{x}^{2}-{5}^{2}}}$$

$$=\dfrac{1}{2\times 5}\log{\left|\dfrac{x-5}{x+5}\right|}+c$$

$$=\dfrac{1}{10}\log{\left|\dfrac{x-5}{x+5}\right|}+c$$

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